If the expression \frac{(x^3 y^2)^3 \cdot (x^{-2} y^{-1})^2}{x^k y^3} is equivalent to x^2… — ACT

If the expression

\frac{(x^3 y^2)^3 \cdot (x^{-2} y^{-1})^2}{x^k y^3}

is equivalent to $x^2 y$ for all non-zero real numbers $x$ and $y$ , what is the value of the exponent $k$ ?

Answer: 3

The value of the exponent

k

is 3.

Applying exponent rules simplifies the numerator of the expression to

x^5 y^4

. Dividing this by the denominator

x^k y^3

yields

x^{5-k} y

. Equating the exponent of

x

to 2 in the target expression

x^2 y

gives

5 - k = 2

, which solves to

k = 3

Apply the power of a power rule

(a^m)^n = a^{m \cdot n}

to simplify each factor in the numerator.

(x^3 y^2)^3 = x^9 y^6

and

(x^{-2} y^{-1})^2 = x^{-4} y^{-2}

To raise a product to a power, raise each factor to that power by multiplying the exponents.

Multiply the two simplified factors in the numerator together by adding the exponents of like bases.

x^9 y^6 \cdot x^{-4} y^{-2} = x^5 y^4

When multiplying exponential expressions with the same base, add their exponents:

a^m \cdot a^n = a^{m+n}

Divide the numerator by the denominator by subtracting the exponents of like bases.

\frac{x^5 y^4}{x^k y^3} = x^{5-k} y

When dividing exponential expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator:

\frac{a^m}{a^n} = a^{m-n}

Set the exponent of

x

in the simplified expression equal to the exponent of

x

in the target expression

x^2 y

, and solve for

k

5 - k = 2 \implies k = 3

For the expressions to be equivalent for all non-zero real numbers, the corresponding exponents of like bases must be equal.

Properties of exponents (power of a power, product of powers, and quotient of powers rules)

Estimated Time:1m 30s

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